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Figurate number
arranged in an equilateral triangle. The triangular lattice representing the n {\displaystyle n} th triangular number contains n {\displaystyle n} rows: the
Triangular_number
Square of a triangular number
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2
Squared_triangular_number
Centered figurate number that represents a triangle with a dot in the center
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other
Centered_triangular_number
Natural number
a prime number and an emirp. It forms a twin prime pair with 1031. 1035 = 32 × 5 × 23. It is a hexagonal number and the 45th triangular number. 1039 is
1000_(number)
Natural number
as the "number of the beast" or "number of (a) man". 666 is the sum of the first thirty-six natural numbers, which makes it a triangular number: ∑ i =
666_(number)
Integer that is both a perfect square and a triangular number
mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words, the
Square_triangular_number
Type of triangular number
n} th triangular number, then the doubly triangular numbers are the numbers of the form T T n {\displaystyle T_{T_{n}}} . The doubly triangular numbers
Doubly_triangular_number
Type of figurate number
properties of oblong, triangular, and square numbers. The number 10 for example, can be arranged as a triangle (see triangular number): But 10 cannot be
Polygonal_number
Natural number
composite number and a happy number. 10 is the smallest noncototient number. Ten is the fourth triangular number and the third tetrahedral number: 10 = 1
10
Natural number
non-trivial square triangular number. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than
36_(number)
Natural number
It is a Lucas–Carmichael number. 2016 = 25 × 32 × 7. It is the second-smallest Erdős–Nicolas numberand a triangular number. 2017 is a sexy prime with
2000_(number)
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns
Pentagonal_number
Product of an integer with itself
cube numbers and triangular numbers). In the real number system, square numbers are non-negative. A non-negative integer is a square number when its square
Square_number
Number equal to the sum of its proper divisors
2^{p-1}(2^{p}-1)} , each even perfect number is the ( 2 p − 1 ) {\displaystyle (2^{p}-1)} -th triangular number (and hence equal to the sum of the integers
Perfect_number
Natural number
hexagonal number, a happy number, and a triangular number. 28 also appears in the Padovan sequence. In algebraic geometry, 28 is the number of bitangents
28_(number)
Natural number
divides the Euclid number 2999# + 1 3003 – triangular number, only number known to appear eight times in Pascal's triangle; no number is known to appear
3000_(number)
Natural number
003 = Smallest 7-digit prime number 1,000,405 = Smallest triangular number with 7 digits and the 1,414th triangular number 1,002,001 = 10012, palindromic
1,000,000
Type of figurate number
number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number,
Hexagonal_number
Natural number
harmonic divisor number 6201 – square pyramidal number 6216 – triangular number 6217 – super-prime 6229 – super-prime 6232 – amicable number with 6368 6241
6000_(number)
Natural number between 89 and 91
the eleventh nontotient (with 50 the fifth). The twelfth triangular number 78 is the only number to have an aliquot sum equal to 90, aside from the square
90_(number)
Natural number
the natural number following 45 and preceding 47. 46 is a composite number, a centered triangular number, and a Wedderburn-Etherington number, and an Erdős–Woods
46_(number)
Natural number
= 7!, superior highly composite number 5041 = 712, centered octagonal number 5050 – triangular number, Kaprekar number, 5051 – Sophie Germain prime 5059
5000_(number)
Natural number
(four thousand) is the natural number following 3999 and preceding 4001. It is a decagonal number. 4005 – triangular number 4007 – safe prime 4010 – magic
4000_(number)
Natural number
expressed as 25 - 1. 31 is a centered triangular number, a centered pentagonal number, and a centered decagonal number. 31 equal temperament is a popular
31_(number)
Thought experiment of infinite sets
) / 2 {\displaystyle (n(n+1))/2} , or the n {\displaystyle n} th triangular number. Those in a coach will be in room ( ( c + n − 1 ) ( c + n ) ) / 2
Hilbert's paradox of the Grand Hotel
Hilbert's_paradox_of_the_Grand_Hotel
Natural number
7000 (seven thousand) is the natural number following 6999 and preceding 7001. 7021 – triangular number 7043 – Sophie Germain prime 7057 – cuban prime
7000_(number)
Natural number
number following 65 and preceding 67. 66 is a sphenic number, a semiperfect number, and a Erdős–Woods number. It is also the 11th triangular number and
66_(number)
Sum of a factorial number and a triangular number
In number theory, a factoriangular number is an integer formed by adding a factorial and a triangular number with the same index. The name is a portmanteau
Factoriangular_number
Natural number
octahedral number; 8119/5741 ≈ √2 8125 – pentagonal pyramidal number 8128 – perfect number, harmonic divisor number, 127th triangular number, 64th hexagonal
8000_(number)
Polyhedral number representing a tetrahedron
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron
Tetrahedral_number
Natural number
octagonal number 9029 – Sophie Germain prime 9041 – super-prime 9045 – triangular number 9059 – Sophie Germain prime 9072 – decagonal number 9077 – Markov
9000_(number)
Natural number
composite number, the 2nd colossally abundant number, the 3rd triangular number, the 4th highly composite number, a pronic number, a congruent number, a harmonic
6
Natural number
Forty-five degrees is half of a right angle (90°). Forty-five is a triangular number: it is the sum of all single-digit decimal digits: 0 + 1 + 2 + 3 +
45_(number)
Natural number
natural number following 230 and preceding 232. 231 is a sphenic number. 231 is the 21st triangular number, a doubly triangular number, a hexagonal number, an
231_(number)
Natural number
(fifty-five) is the natural number following 54 and preceding 56. 55 is composite number, a Fibonacci number and a Triangular number. In the United States,
55_(number)
Natural number
smallest triangular number divisible by 1000 195,025 = Pell number, Markov number 196,418 = Fibonacci number, Markov number 196,560 = the kissing number in
100,000
Natural number
triangle (as all triangular and tetragonal numbers appear in it). Because 15 is also triangular, 120 is a doubly triangular number. 120 is divisible
120_(number)
Natural number
natural number following 299 and preceding 301. 300 is a composite number and the 24th triangular number. It is also a second hexagonal number. 303 is
300_(number)
Natural number
It is a centered triangular number and a happy number. 902 = 2 × 11 × 41. It is a sphenic number, a nontotient, and a Harshad number. 903 = 3 × 7 × 43
900_(number)
Numbers arranged in a triangle
parentheses with a given number of distinct nestings Pascal's triangle, whose entries are the binomial coefficients Clark's triangle Triangular arrays of integers
Triangular_array
Figurate number
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers,
Pyramidal_number
Natural number
second and only prime triangular number, and Carl Friedrich Gauss proved that every integer is the sum of at most three triangular numbers. Three is the
3
Natural number
019 = Smallest 8-digit prime number 10,001,628 = Smallest triangular number with 8 digits and the 4,472nd triangular number 10,004,569 = 31632, the smallest
10,000,000
Natural number
to 38, or twice 19. A hexaflexagon is a strip of nineteen alternating triangular faces that can flex into a regular hexagon, such that any two of six colorings
19_(number)
Number used to approximate the square root of 2
identity describes a square number, while the right side describes a triangular number, so the result is a square triangular number. Falcón and Díaz-Barrero
Pell_number
Natural number
is the natural number following 165 and preceding 167. 166 is an even number and a composite number. It is a centered triangular number. Given 166, the
166_(number)
Natural number
007 : smallest prime number with 10 digits. 1,000,006,281 : smallest triangular number with 10 digits and the 44,721st triangular number. 1,000,014,129 =
1,000,000,000
Natural number
number 15 is the magic constant of the unique order-3 normal magic square. 15 is the first number to be polygonal in 3 ways: it is the 5th triangular
15_(number)
Natural number
successive triangular numbers 55 and 66. There are 605 non-isomorphic set-systems of weight 9. 606 = 2 × 3 × 101. It is a sphenic number, an admirable number and
600_(number)
Natural number
23,1,0). an octahedral number. a centered triangular number. a centered square number. a decagonal number. the smallest number that can be expressed as
85_(number)
Natural number
prime number 10,000,002,437,316 : smallest 14-digit triangular number, 4,472,136th triangular number 10,153,507,819,457 : 177th Markov number 10,445
10,000,000,000,000
Natural number
a natural number and integer following 152 and preceding 154. It is the sum of the first 17 integers (making it the 17th triangular number), and also
153_(number)
Natural number
pronic number, a nontotient, and a Harshad number. 703 = 19 × 37. It is a hexagonal number, a Kaprekar number and the 37th triangular number. It is the
700_(number)
Natural number
figurate number it is the 23rd triangular number, a hexagonal number, and a centered pentagonal number, the third number after 1 and 6 to have this combination
276_(number)
Natural number
equivalently the 36th triangular number. Furthermore, the sum between twin primes (149, 151) is 300, which in turn is the 24th triangular number. "Sloane's A062786 :
151_(number)
Natural number
N. J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
9
Natural number
29. It is a sphenic number, the 28th triangular number, a centered nonagonal number, an even nontotient, and a Narayana's cow number. English Wikisource
400_(number)
Natural number
hypercomplex number system. 32 is the ninth 10-happy number, while 23 is the sixth. Their sum is 55, which is the tenth triangular number, while their
32_(number)
Natural number
91 is the fourth composite number in the 11-aliquot tree. (91, 51, 21, 18). the 13th triangular number. a hexagonal number, one of the few such numbers
91_(number)
Natural number
is the natural number following 135 and preceding 137. 136 is: a refactorable number and a composite number. the 16th triangular number. a repdigit in
136_(number)
Natural number
number following 189 and preceding 191. 190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after
190_(number)
Natural number
3-aliquot tree. a semiperfect number, as a multiple of a perfect number. the 12th triangular number. a palindromic number in bases 5 (3035), 7 (1417),
78_(number)
Natural number
a happy number, the sum of eight consecutive primes (83 + 89 + 97 + 101 + 103 + 107 + 109 + 113), and the sum of 4 consecutive triangular numbers (171
800_(number)
Natural number
both its prime factors being Gaussian primes. While 21 is the sixth triangular number, it is also the sum of the divisors of the first five positive integers:
21_(number)
Natural number
smallest nine digit prime 100,005,153 = smallest triangular number with 9 digits and the 14,142nd triangular number 100,020,001 = 100012, palindromic square 100
100,000,000
Counts pieces of a disk cut by lines
{n}{1}}+{\dbinom {n}{2}}.} Simply put, each number equals a triangular number plus 1. These are the first number on each row of Floyd's triangle. As the third
Lazy_caterer's_sequence
Number associated with the Beast of Revelation
because it is a triangular number, the sum of the first 36 numbers (1+2+3+4+5+6+...+36 = 666)". The beast's identity and the beast's number are usually interpreted
Number_of_the_beast
Natural number
It is the 32nd triangular number, and the 167th Totient number. 529 = 232. It is a centered octagonal number and a lazy caterer number. 530 = 2 × 5 ×
500_(number)
Natural number
N. J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
100
Natural number
previous prime is 107, making them both twin primes. 109 is a centered triangular number. There are exactly: 109 different families of subsets of a three-element
109_(number)
Natural number
the number of pairs of 23 items and 22nd triangular number, 253. 23 is the smallest discriminant of imaginary quadratic fields with class number 3 (negated)
23_(number)
Centered figurate number that represents a decagon with a dot in the center
other centered k-gonal number, the nth centered decagonal number can be reckoned by multiplying the (n − 1)th triangular number by k, 10 in this case,
Centered_decagonal_number
Size of a geometric arrangement of points
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes
Figurate_number
Natural number
the fourth square triangular number. As a figurate number, 204 is also a nonagonal number and a truncated triangular pyramid number. 204 is a member of
204_(number)
Natural number
preceding 236. 235 is a heptagonal number, a centered triangular number, and a Smarandache–Wellin number. It is a Harshad number in bases 6, 47, 48, 95, 116
235_(number)
Natural number
to 216 can be written as either a triangular number, a prime number or as a triangular number plus a prime number; however, this is not possible for
216_(number)
Class of series of figurate numbers, each having a central dot
k-gonal number contains k more dots than the previous layer. Each centered k-gonal number in the series is k times the previous triangular number, plus
Centered_polygonal_number
Natural number
sum to 496. It is also a triangular and a hexagonal number. and a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample
496_(number)
Symbol of ten points laid in four rows
fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was
Tetractys
Natural number
prime number: 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210. It is the 20th triangular number (following 190 and preceding 231), a pentagonal number (following
210_(number)
Two-dimensional packing problem
triangular number of circles, and conjectures are available for n ≤ 34. A conjecture of Paul Erdős and Norman Oler states that, if n is a triangular number
Circle packing in an equilateral triangle
Circle_packing_in_an_equilateral_triangle
Centered figurate number that represents a nonagon with a dot in the center
simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number. Thus, the first few
Centered_nonagonal_number
Natural number
smallest 11-digit prime number. 10,000,020,331 = smallest triangular number with 11 digits and the 141,421st triangular number 10,123,457,689 = smallest
10,000,000,000
Natural number
octahedral number 11781 = triangular number, hexagonal number, octagonal number, and also 58-gonal, 216-gonal, 329-gonal, 787-gonal and 3928-gonal number 11811
10,000
Natural number
ways. It is a square number (225 = 152), an octagonal number, and a squared triangular number (225 = (1 + 2 + 3 + 4 + 5)2 = 13 + 23 + 33 + 43 + 53)
225_(number)
Natural number
same amount of digits the 22nd triangular number. a star number. a centered heptagonal number. a centered nonagonal number. a Blum integer. a member of
253_(number)
Natural number
smallest 12-digit prime number. 100,000,147,984 = 3162282, the smallest 12-digit square. 100,000,404,505 = smallest triangular number with 12 digits and the
100,000,000,000
Natural number
4, 25, 196, ... in which each number is the smallest square that differs from the previous number by a triangular number. There are 196 one-sided heptominoes
196_(number)
Constant used in a magic square
which is also a: triangular number is 15 (solve the Diophantine equation x2 = y3 + 16y + 16, where y is divisible by 4); square number is 1 (solve the
Magic_constant
Natural number
pyramidal number and a dodecagonal number. Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number: 41616
288_(number)
Number, product of consecutive integers
triangular number and n more than the nth square number, as given by the alternative formula n2 + n for pronic numbers. Hence the nth pronic number and
Pronic_number
Number that represents a hexagon with a dot in the center
{n(n-1)}{2}}\right)} shows that the centered hexagonal number for n is 1 more than 6 times the (n − 1)th triangular number. In the opposite direction, the index n corresponding
Centered_hexagonal_number
Number in the 5th cell of any row of Pascal's triangle
In number theory, a pentatope number (or hypertetrahedral number or triangulo-triangular number) is a number in the fifth cell of any row of Pascal's
Pentatope_number
numbers of the second kind Triangular number Triangular pyramidal number The (incomplete) Bell polynomials from a triangular array of polynomials (see
List_of_triangle_topics
Natural number
81 − 8 ) {\displaystyle 4^{3}+3^{4}-2^{3}=(64+81-8)} , cube of 4 + Triangular number T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as
137_(number)
Natural number
untouchable number, a triangular number, hexagonal number, 100-gonal number, and is polygonal in 6 other ways. 61,871 = approximate number of planck lengths
60,000
Natural number
seventy-one) is the natural number following 170 and preceding 172. 171 is the 18th triangular number and a Jacobsthal number. There are 171 transitive
171_(number)
Chapter of the New Testament
significance of the number (which will be developed further in chapter 17). The number 666 is unusual for being a triangular number (that is, the sum of
Revelation_13
Number raised to the third power
For example, the sum of the first 5 cubes is the square of the 5th triangular number, 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 15 2 {\displaystyle
Cube_(algebra)
Natural number
prime number 100,000,012,392,316 : smallest 15-digit triangular number, 14,142,136th triangular number 100,169,256,075,517 : 202nd Markov number 106,974
100,000,000,000,000
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